Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial.

Definition. A square principle for a cardinal $\theta$ asserts the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\theta ⟩$ such that for all $\alpha <\theta$, $C_{\alpha }$ is a closed subset of $\theta$.

Characteristics

1. $S(\mathrm{\Gamma })$, the support of $\mathrm{\Gamma }$, is the set $\{\alpha <\theta \mid C_{\alpha }\subseteq sup(C_{\alpha })=\alpha \}$;
2. $w(\mathrm{\Gamma })$, the width of $\mathrm{\Gamma }$, is the cardinal $sup\{|\mathcal{T}(\mathrm{\Gamma },\alpha )|\mid \alpha <\theta \}$, where $$\mathcal{T}(\mathrm{\Gamma },\alpha )=\{C_{\beta }\cap \alpha \mid \beta <\theta ,sup(C_{\beta }\cap \alpha )=\alpha \};$$
3. Nontriviality condition.

Examples:

(Jensen, 1972). In $L$, for every infinite cardinal $\kappa$, ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\kappa }$ holds. That is, there exists a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <{\kappa }^{+}⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <{\kappa }^{+}\mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })=1$;
3. $\text{otp}(C_{\alpha })\le \kappa$ for all $\alpha <{\kappa }^{+}$.

(Jensen, 1972). In $L$, if $\kappa$ is a regular uncountable cardinal that is not weakly compact, then there exists a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\kappa ⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <\kappa \mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })=1$;
3. there exists a stationary subset $E\subset \kappa$ such that $E$ contains no accumulation points of $C_{\alpha }$ for all $\alpha <\kappa$.

(Todorcevic, 1984). PFA refutes the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\theta ⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <\theta \mid \text{cf}(\alpha )={\omega }_1\}$, $\theta =\text{cf}(\theta )>{\omega }_1$;
2. $w(\mathrm{\Gamma })=1$;
3. there exists no club $C$ in $\theta$ such that $C\cap \alpha =C_{\alpha }$ for every $\alpha \in S(\mathrm{\Gamma })$ which is an accumulation point of $C$.

(Baumgartner, 198?, Cummings-Magidor, 2011, Brodsky-Rinot, 2016). PFA is consistent with ${\boxminus }_{\kappa ,\ge {\mathrm{\aleph }}_2}$ for every singular cardinal $\kappa$, where ${\boxminus }_{\kappa ,\ge \chi }$ asserts the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <{\kappa }^{+}⟩$ with

1. $S(\mathrm{\Gamma })\supseteq \{\alpha <{\kappa }^{+}\mid \text{cf}(\alpha )\ge \chi \}$;
2. $w(\mathrm{\Gamma })=1$;
3. $\text{otp}(C_{\alpha })\le \kappa$ for all $\alpha <{\kappa }^{+}$.

(Velickovic, 1986, Todorcevic, 1987, Shelah-Stanley, 1988). If $\theta =\text{cf}(\theta )>{\omega }_1$ is not weakly compact in L, then $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}(\theta )$ holds. That is, there exists a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\theta ⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <\theta \mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })=1$;
3. there exists no club $C$ in $\theta$ such that $C\cap \alpha =C_{\alpha }$ for every $\alpha \in S(\mathrm{\Gamma })$ which is an accumulation point of $C$.

(the proof in Velickovic, 1986 focuses on the construction of $C_{\alpha }$ for $\alpha$ which is a regular cardinal. The proof in Todorcevic, 1987 is more detailed, and the proof in Shelah-Stanley, 1988 is the most detailed.)

(Rinot, 2016). If GCH holds and $\theta >{\omega }_1$ is a successor cardinal which is not weakly compact in L, then ${⊠}^{-}(\theta )$ holds. That is, there exists a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\theta ⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <\theta \mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })=1$;
3. for every cofinal subset $A\subseteq \theta$, there exists some nonzero limit ordinal $\alpha <\kappa$ such that $sup(\text{nacc}(C_{\alpha })\cap A)=\alpha$.

The fact that ${⊠}^{-}(\theta )$ implies $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}(\theta )$ may be found in here.

(Shelah, 1991. See here). If $\lambda <\kappa$ are regular cardinals, and $S$ is a stationary subset of $E_{\lambda }^{{\kappa }^{+}}$, then there exists a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <{\kappa }^{+}⟩$ with

1. $S(\mathrm{\Gamma })\cap S$ is stationary;
2. $w(\mathrm{\Gamma })=1$;
3. $\text{otp}(C_{\alpha })\le \lambda$ for all $\alpha <{\kappa }^{+}$.

(Schimmerling, 1995). ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\kappa ,\lambda }$ asserts the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <{\kappa }^{+}⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <{\kappa }^{+}\mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })\le \lambda$;
3. $\text{otp}(C_{\alpha })\le \kappa$ for all $\alpha <{\kappa }^{+}$.

(Krueger, 2013, extending Jensen, 1972). Weak square on $\kappa$ asserts the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\kappa ⟩$ with

1. $S(\mathrm{\Gamma })$ is a club in $\kappa$ consisting of singular ordinals (so $\kappa$ is non-Mahlo);
2. $|\mathcal{T}(\mathrm{\Gamma },\alpha )|<\kappa$ for all $\alpha <\kappa$;
3. $\text{otp}(C_{\alpha })<\alpha$ for all $\alpha <\kappa$.

(Todorcevic and Torres Perez, 2014). Rado’s conjecture refutes the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <\theta ⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <\theta \mid \alpha \text{ limit}\}$, $\theta =\text{cf}(\theta )>{\omega }_1$, not the successor of a singular cardinal of countably cofinality;
2. $w(\mathrm{\Gamma })\le {\mathrm{\aleph }}_0$;
3. there exists no sequence $⟨A_{\alpha }\mid \alpha <\theta ⟩$ of countable subsets of $\theta$ satisfying the following. For all $\alpha <\beta <\theta$, there exist $\tau \in S(\mathrm{\Gamma })$ and $\gamma <\delta$ with $(\gamma ,\delta )\in A_{\alpha }\times A_{\beta }$ such that $sup(C_{\tau }\cap \gamma )=\gamma$, $sup(C_{\tau }\cap \delta )=\delta$.

(Neeman, 2014). ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\kappa ,\lambda }^{\text{ta}}$ asserts the existence of a sequence $\mathrm{\Gamma }=⟨C_{\alpha }\mid \alpha <{\kappa }^{+}⟩$ with

1. $S(\mathrm{\Gamma })=\{\alpha <{\kappa }^{+}\mid \alpha \text{ limit}\}$;
2. $w(\mathrm{\Gamma })\le \lambda$;
3. $\text{otp}(C_{\alpha })<\kappa$ for all $\alpha <{\kappa }^{+}$ of cofinality $<\kappa$, and for all $C,D\in \mathcal{T}(\mathrm{\Gamma },\alpha )$, there exists $\beta <\alpha$ with $C\setminus \beta =D\setminus \beta$.

For more square principles, see the papers by Cummings and Schimmerling, Krueger and Schimmerling, and Brodsky and Rinot. The following presentation by Hiroshi Sakai presents the state-of-the-art of the study of failure of these principles.
Foreman and Magidor wrote an interesting paper on a couple of weak forms of square, but these principles fit more into the so-called approachability principles. What is the difference between square principles and approachability principles? the major ones are that the sets $C_{\alpha }$ do not have to be closed, and that instead of measuring the width of $\mathrm{\Gamma }$, one seeks to occur into (some or all) bounded subsets of $C_{\alpha }$ sometime before arriving to $\alpha$.
An important paper on approachability principles is the one by Dzamonja and Shelah. Finally, it may worth metnioning that in Section 4 of this paper, I am constructing a model where (weak) square fails at ${\mathrm{\aleph }}_{\omega }$, yet a (quite useful) approachability principle holds.